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Triangles

Long Answer Type

381. ABCD is a parallelogram, P is a point on side BC and DP when produced meets AB produced in L.
Prove that: (i)

DP over PL equals DC over BL

(ii)

DL over DP space equals space AL over DC.

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382. If A is the area of a right-angle triangle and b is one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is $fraction numerator 2 Ab over denominator square root of straight b squared plus 4 straight A to the power of 4 end root end fraction.$

Solution not provided.

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Short Answer Type

383.

In the Fig, AB || DE and BD || EF. Prove that DC² = CF × AC.

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384. The perpendicular AD on the base BC of ∆ABC meets BC at D, such that BD = 3 CD. Prove that 2(AB² - AC²) = BC².

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385.

In the given Fig, AD ⊥ BC if D divides BC in the ratio 1 : 3 prove that: 2 AC² = 2AB² + BC².

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386. In a triangle ABC, ∠C = 90° and AQ is a median. Prove that: BC² = 4 (AQ² - AC²).

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387. The side QR of a square PQRS is produced to T. Prove that: PT² = RT² + 2QR × QT.

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Long Answer Type

388. In Fig, AD is the median of ∆ABC and AE ⊥ BC. If BC = a, CA = b, AD = p, AE = h and DE = x. Prove that

(i)

straight b squared space equals space straight p squared plus ax plus straight a squared over 4.

(ii)

straight c squared equals straight p squared minus ax plus straight a squared over 4.

(iii)

straight b squared plus straight c squared equals 2 straight p squared plus straight a squared over 2.

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Short Answer Type

389. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the lenght of her shadow after 4 seconds.

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390. Two right triangles ∆PQR and ∆SQR are drawn on the same side of the hypotenuse QR. If PR and SQ intersect at M such that PM = 3cm, MR = 6cm and SM = 4 cm, find MQ.

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